How can I fix a melted spot on a plastic sink?

If it's Corian, some damage can be sanded out. Check the Corian website. (Don't know about that big burn, though.) If you decide to replace it, go with a porcelain or stainless steel, if possible, and get your daughter a large trivet for her curling iron--that's what I did.

You might try heating up the curling iron again to try and smooth that big crease out and then lightly sand with a really fine sand paper and then buff with car polish the cutting kind, otherwise it is pretty much looking like a lost cause any way.

I would use the flat iron and make shapes into the sink for a new look. If you do it right, everyone will be wanting one just like yours.

I just watched a video fixing chips in helicopter blades. They used super glue and baking soda. Fill the spot with super glue (may take several fills) and immediately pour baking soda on it. It will harden immediately. If I can find that video I'll post the link...https://www.youtube.com/watch?v=KU2qph2vG9Y

GET A NEW ONE

1) You could smooth the edges of the melted areas and fill then in with Automotive "Spot Putty" which is usually white to off white. If the color match isn't close enough tape off the surrounding area and faucet and drain; remove the valve knobs and apply 3-4 coats of spray paint for plastics allowing enough drying time between coats. When removing the high spots on the edges use a very fine single cut file 8"-10" long or very fine sand paper. Every scratch made will show up in the finished product.

2) Instill in her that she has to be more careful and make her look up the replacement cost.

3) Buy a new sink and make her help replace it.

Bondo?

https://www.hunker.com/13423684/how-to-remove-burn-marks-from-plastic-bathroom-sinks

Sand out with 150, 200 and finish with 400 grit sand paper. Final sanding with 400 grit would be better wet sanded. Finish up the spots by polishing with automotive rubbing compound and and apply a coat of automotive wax to seal it